After the sprint, students should take one or two minutes to consider strategies and to make corrections to the sprint.

Then the students should do logarithm sprint 4. After this sprint it's a good idea to poll the class for their impressions. Are they improving? Do they have any strategies for thinking about these problems?

What your students should be noticing is the inverse relationship between logarithms and exponents. The sprints are designed to encourage the kind of thought process that sees something like and translates it to . Since the students are so familiar with exponentiation, logarithms become easy when viewed this way. I emphasize that this is a matter of translating from the language of exponents to the language of logarithms, and vice versa.

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As they begin to work individually or in small groups, I will take a quick walk around the room to get a sense of how much everyone has left. I expect that most students will be nearly finished, but I also expect them to be slowed down by the the final two problems.

A common question is how many decimal places are required for a "decent" approximation. I usually suggest that they find a value of x that brings them within one one-thousandth of the correct power. For 4^x = 95, this means they should find x = 3.285.

For the final change-of-base problems, I offer the suggestion that students consider rewriting the expressions in this way: . From here, they are typically able to use the properties of exponents to solve for "?". The final question presents an additional challenge because the two bases are not integer powers of the same number. I have included it as a challenge for my best students, and I am surprised every year at how many of them are able to figure it out!

(Incidentally, the change-of-base formula goes beyond what is required by the CCSSM, but I think it's important to include it since students always ask how to evaluate logs in bases other than 10 or e. That said, many calculators today are able to evaluate logarithms in any base, so the motivation to teach this property at this stage may be gone before too long.)

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The groundwork for the properties of logarithms has been laid with the problems that the students just completed. Before going further, we'll correct any mistakes by asking individual students to share their answers with the class.

Along the way, I'll ask students to try to generalize any patterns they're seeing or using. I will guide them toward the first properties of logarithms, and I might express them on the board like this:

(log of factor) + (log of factor) = (log of product)

(log of product) - (log of factor) = (log of factor)

For the time being, I'll leave it at that and move on, but I'll be sure to keep these on the board for later reference.

As we check our solutions to approximation problems, I'll use this opportunity to reiterate the convenience of agreeing on a "common logarithm", one with a base of 10. I'll also use this chance to bring up the natural logarithm. Many students have noticed the "ln" button on their calculators by now, as well as its proximity to "e^x", so they guess that it's a logarithm of some kind. I'll point it out to everyone now and tell them that this kind of logarithm will become very important in the future. My students may have never heard of the number e, but I'll assure them that we'll get to know it better later. (If your students are already familiar with e, you would naturally want to go a little deeper with this.)

Finally, most of the conversion problems require nothing more than the properties of exponents, but the last question is much more challenging. There are a number of ways to approach it, but I would probably guide them toward thinking that . From here, they can solve the problem just like the others, but in terms of a logarithm.

Now, it's time to refer back to that expression we wrote on the board earlier:

(log of factor) + (log of factor) = (log of product).

First, we'll express this more formally with symbolic notation. This is our first property of logarithms, but why does it work?

The first hint I'll give is to remind the class that logarithms are exponents, so this must really be a property of exponents. Of course, this first hint may not be enough, so I'll slowly take my students through the thought process outlined here. My goal is to show that this is not a new property of a new mathematical object at all! It's simply a translation of a familiar fact into the language of logarithms!